2 Spin is quantizedFor electrons (& other fermions), spin can only take on two values: up ↑ or down ↓.What’s so special about the z-axis?Answer: nothing.Can measure spin along any axis, will always find spin either aligned or anti-aligned with the axis you measure along.Just like position and momentum, spin along orthogonal axes obeys Heisenberg uncertainty principle: sxsz≥/2; sysz≥/2; sxsy≥/2State of definite spin in x-direction -->50/50 superposition of up and down in z-direction.

3 z z How do we know? Stern-Gerlach ExperimentPut atoms in inhomogeneous magnetic field pointing inz direction – split in two groups – spin up and spin downzWhat if I take just atoms that went up, and send themthrough another, identical magnetic field – What happens?zHalf go up (+z), half go down (–z)All go up (+z)All go down (–z)Range of paths all smeared outMagnetic field measures spin of atoms. All atoms that end up on top are in definite state of spin up.

4 z x Second Experiment: What if I take just atoms that went up, andsend them through a magnetic field pointed in the x direction– perpendicular to first field (pointing into the screen)?zxHalf go into the screen (+x), half go out of the screen (–x)All go into the screen (+x)All go straight (no deflection)Range of paths all smeared outAll go up (+z)Spin in x direction and z direction are “incompatible observables” – like position and momentum. Obey uncertainty relation sxsz≥/2. Can’t know both at same time!

5 Third Experiment: Take just the atoms that went in +x direction insecond experiment, and send them through a third magnetic field,pointed in the z direction?zxzHalf go up (+z), half go down (–z).All go up (+z)All go down (–z)Range of paths all smeared out.Even though all atoms going into final field started out in state +sz, they are now in state +sx, spin x measurement destroyed spin z info.